8th Swedish Mathematical Society Problems 1968

1.  Find the maximum and minimum values of x² + 2y² + 3z² for real x, y, z satisfying x² + y² + z² = 1.

2.  How many different ways (up to rotation) are there of labeling the faces of a cube with the numbers 1, 2, ... , 6?

3.  Show that the sum of the squares of the sides of a quadrilateral is at least the sum of the squares of the diagonals. When does equality hold?

4.  For n ≠ 0, let f(n) be the largest k such that 3^k divides n. If M is a set of n > 1 integers, show that the number of possible values for f(m-n), where m, n belong to M cannot exceed n-1.

5.  Let a, b be non-zero integers. Let m(a, b) be the smallest value of cos ax + cos bx (for real x). Show that for some r, m(a, b) ≤ r < 0 for all a, b. 

Solutions

Problem 1
Find the maximum and minimum values of x² + 2y² + 3z² for real x, y, z satisfying x² + y² + z² = 1.
Answer
3, 1
Solution

x² + 2y² + 3z² = (x² + y² + z²) + y² + 2z² = 1 + y² + 2z² ≥ 1. Equality if x = 1, y = z = 0.

x² + 2y² + 3z² ≤ 3(x² + y² + z²) = 3. Equality if x = y = 0, z = 1.

Problem 2

How many different ways (up to rotation) are there of labeling the faces of a cube with the numbers 1, 2, ... , 6?

Answer

30

Solution

Rotate the cube so that 1 is uppermost. There are then 5 choices for the opposite face. There are then 4! = 24 ways of arranging the other 4 faces, but groups of 4 are related by rotation, so 5·6 = 30 ways in all.